Tutorial: Complexity of Many-Valued Logics

Like in the case of classical logic and other non-standard logics, a variety of complexity-related questions can be asked in the context of many-valued logic. Some questions, such as the complexity of the sets of satisfiable and valid formulas in various logics, are completely standard; others, such as the maximal size of representations of many-valued connectives, only make sense in a many-valued context. In this overview I concentrate mainly on two kinds of complexity problems related to many-valued logics: in Section 6 I discuss the complexity of the membership problem in various languages, such as the satisfiable, respectively, the valid formulas in some well-known logics. Two basic proof techniques are presented in some detail: a reduction of many-valued logic to mixed integer programming and a reduction to classical logic. I do not mention results on complexity of algorithms, in particular, of constructing decision diagrams, because they are a quickly moving target. Instead, in Section 7, I discuss the size of representations of many-valued connectives and quantifiers, because this has a direct impact on the complexity of many kinds of deduction systems. I consider results on both propositional and firstorder logic throughout the article. The following Section 2 sets up a framework, in which a wide class of many-valued logics can be discussed. This is followed by three brief sections that introduce into Signed Logics, Mixed Integer Programming, and Polynomial Expressions, respectively. I tried to make this article self-contained.